(iii) Theoretical Geomagnetism
Lecture 8:
Fundamentals of Electromagnetism
and the Induction equation
8.0 Introduction
• In earlier lectures we have established that:
(i) Earth’s outer core consists of liquid metal (mostly Fe).
(ii) The majority of the geomagnetic field originates in the core.
(iii) This magnetic field changes on timescales of 1 up to 108 yrs.
• It seems likely that it is the motion of liquid metal in the core that
both generates the geomagnetic field and causes its time variations.
But, how does this happen in detail?
To answer we need to understand the physics of electromagnetism
applied to moving conductors..........
Lecture 8: Fundamentals of Electromagnetism
and the Induction equation
8.1 Induction by a moving conductor
8.2 Maxwell’s equations of Electromagnetism
8.3 Approximations and moving reference frames
8.4 The Induction Equation
8.5 Summary
Appendix: Ohm’s law from microscopic considerations
8.1 Induction by a moving conductor
• To begin, let us consider what happens when an electrical conductor
is pulled through a magnetic field:
• 3 effects result:
(From Davidson, 2001)
(1) An electrical current is induced in the conductor.
(2) This current causes a magnetic field that adds to the original field,
such that the conductor appears to drag the field along with it.
(3) The combined magnetic field interacts with the current resulting in a
Lorentz force that acts on conductor, opposing its motion.
Lecture 8: Fundamentals of Electromagnetism
and the Induction equation
8.1 Induction by a moving conductor
8.2 Maxwell’s equations of Electromagnetism
8.3 Approximations and moving reference frames
8.4 The Induction Equation
8.5 Summary
Appendix Ohm’s law from microscopic considerations
8.2.1 Basic Assumptions
• Our mathematical description of electromagnetism will be in terms of
macroscopic, continuum, physical quantities measured in SI units:
B(r, θ, φ, t)
Magnetic Flux Density (often called Magnetic Field) in Tesla (T).
E(r, θ, φ, t)
Electric Field in Volts per metre (Vm-1).
J (r, θ, φ, t)
Electric Current Density in Amperes per cubic metre (Am-3)
ρe (r, θ, φ, t)
Electric Charge Density in Coulombs per cubic metre (Cm-3)
i.e. It is assumed that these fields are well defined and smoothly varying
everywhere in space and individual particles are ignored.
(N.B. Good approximation if considering length scales much larger than
the size and mean free path length of the particles in the medium.)
8.2.2 Maxwell’s equations: An Overview
• The relationship between magnetic fields, electric fields and
electrical currents are encapsulated in just 4 simple equations:
ρe
∇·E =
"0
∇·B =0
∂B
∇×E =−
∂t
Gauss’s Law: E- fields produced by charge density
Lack of monopole sources of B-field so B-field is solenoidal.
Faraday’s Law of Induction: E-field induced by changing B-field
∂E
∇ × B = µ0 J + !0 µ0
∂t
Ampere-Maxwell law: B-field produced by currents
or by changing E-field(displacement currents)
• Note, this is the form of Maxwell’s equations for materials with
no permanent magnetization or electric polarization and where
!0 = electrical permittivity of free space = 8.85 x10
µ0 = magnetic permeability of free space = 4! x
-12
10-7
C2 N-1 m-2
N
A-2
(!0 µ0 )−1/2 = c = 3 × 108 ms−1
and
8.2.3 Maxwell’s equations:
Gauss’s law of electrostatics
• Consider the Gauss’s law
∇·E =
ρe
"0
• If there is no time-varying magnetic field, then by Faraday’s Law
∇×E =0
and
E = −∇φ
• The electric potential is then defined by Poisson’s equation,
ρe
#0 !
1
ρe (r ! ) 3 ,
φ(r) =
d r
4π#0
|r − r ! |
∇2 φ = −
with solutions
of the form:
• Thus φ and hence E at any point can be obtained by integration
over all the sources of electric field (the charge density).
• Considering a single charge at the origin and noting that the force on
another charge is F = qE, gives Coulomb’s law of electrostatics.
8.2.4 Maxwell’s equations:
Divergence-free Magnetic fields
• Consider the Maxwell equation,
∇·B =0
• This expresses the observational fact that there are no point
sources/sinks of magnetic field (i.e. no magnetic monopoles)
• Therefore:
(i) Magnetic field lines can never end, but are always closed
(ii) Same number of field lines as enter must leave a volume
• The continuity equation for incompressible fluids (∇ · u = 0)
takes the same form, so the same amount of fluid as
enters a volume must leave a volume.
8.2.5 Maxwell’s equations:
Faraday’s Law of EM Induction
• The differential form of Faraday’s Law may be integrated over a
surface S perpendicular to the magnetic field,
!
(∇ × E) · dS = −
S
!
S
∂B
· dS
∂t
• Then using Stokes Theorem the LHS can be re-written as a line
integral round the curve C surrounding S such that,
!
dΦ
E · dl = −
dt
C
where Φ =
!
B · dS
S
• Therefore, the change in magnetic flux through S is equal to the
induced electric field integrated round C.
8.2.6 Maxwell’s equations:
Ampere-Maxwell Law
• Assuming that changes in the
electric field are slow then,
• Integrating this over a surface S
perpendicular to the B-field,
∇ × B = µ0 J
!
(∇ × B) · dS = µ0
!
J · dS
S
S
• Then using Stokes theorem !
B · dl = µ0 I
on the LHS gives us
C
Ampere’s law:
where
I=
!
J · dS
S
• If we instead take the curl of ∇ × B = µ0 J we obtain a vectorial
Poisson equation:
2
∇ B = −µ0 ∇ × J
• The solution of this equation is known as the Biot-Savart Law,
!
!
µ0
µ0
J (r ! ) × (r − r ! ) 3 !
J (r ! ) 3 !
d r
B(r) =
∇×
d r =
4π
|r − r ! |
4π
|r − r ! |3
Lecture 8: Fundamentals of Electromagnetism
and the Induction equation
8.1 Induction by a moving conductor
8.2 Maxwell’s equations of Electromagnetism
8.3 Approximations and moving reference frames
8.5 The Induction Equation
8.6 Summary
Appendix: Ohm’s law from microscopic considerations
8.3.1 Approximations:
Neglect of the displacement current
• Consider the relative magnitude of the 2nd term on the RHS and
the term on the LHS of the Ampere-Maxwell equation:
!0 µ0 E/τ
|!0 µ0 ∂E/∂t|
∼
|∇ × B|
B/l
• From similar scale analysis of terms in the Faraday’s Law equation
we have, |∇ × E| ∼ | − ∂B/∂t| so that E/l ∼ B/τ therefore
1 l2
|!0 µ0 ∂E/∂t|
∼ 2 2
|∇ × B|
c τ
8.3.1 Approximations:
Neglect of the displacement current
• l, τ are the characteristic length and time scales associated with
EM field changes we wish to study. For Earth’s core, the largest
lengthscales are ~ 3481km and shortest timescales ~1 yr
1 l2
|!0 µ0 ∂E/∂t|
∼ 2 2
|∇ × B|
c τ
∼
!
0.1ms−1
3 × 108 ms−1
"2
-19
∼ 10−20
• Therefore can safely neglect the displacement current term:
Physically this represents filtering out EM waves from the system.
8.3.2 Moving reference frames:
The Lorentz transformations
• The observed electric and magnetic field depend on the frame of
reference in which they are measured (Einstein, 1905).
!
• Let E and B be the fields in one frame of reference and E , B
be those measured in a second inertial frame moving at velocity
u relative to the first. The transformation btw the frames is:
!
(u · E)u
u2
(u · B)u
B ! = γ(B + "0 µ0 (u × E)) + (1 − γ)
u2
E ! = γ(E + u × B) + (1 − γ)
where γ = (1 − u2 /c2 )−1/2
8.3.2 Moving reference frames:
The Lorentz transformations
(u · E)u
u2
(u · B)u
B ! = γ(B + "0 µ0 (u × E)) + (1 − γ)
u2
E ! = γ(E + u × B) + (1 − γ)
• Assuming motions in Earth’s core are non-relativistic i.e. u2 << c2
2 2 −1/2
then γ = (1 − u /c )
~ 1 and the transformations simplify to
E! = E + u × B
and B ! = B
• Thus moving to a reference frame travelling along with the core
fluid: (i) the magnetic field is unchanged, (ii) the electric field is
modified by effect of magnetic field on moving charged particles.
8.3.3 Ohm’s law in a moving reference frame
• Ohm’s law is an empirical law stating the experimentally observed
relation between the electric fields and electric current density.
• For stationary conductors it takes the form:
J = σE
where
σ = electrical conductivity ( Ω−1 m−1 )
• When considering a moving electrical conductor, the effective
electric field in the frame moving with the conductor must be used:
J = σE ! = σ(E + u × B)
• Again, note, this a non-relativistic approximation ( u2 << c2 ).
• For Earth’s core (predominantly liquid Fe at high P,T) the electrical
conductivity is thought to be large ~ 0.5x106 Ω−1 m−1
8.3.4 Summary of Equations describing
Electrodynamics
of 2the Core
2
• For moving conductors (where u << c ) and considering
slow changes in the EM fields ( (l/τ )2 /c2 << 1 ) then the evolution
of the magnetic fields and electric currents are specified by:
∇·B =0
∂B
∂t
∇ × B = µ0 J
∇×E =−
J = σ(E + u × B)
• N.B.1: Only time-derivative of the magnetic field remains.
• N.B.2: Neglect of the displacement current means we no longer
need to consider the Gauss’s electrostatic eqn (decoupled).
Often referred to as:
“the magnetohydrodynamic (MHD) approximation of electrodynamics”
Lecture 8: Fundamentals of Electromagnetism
and the Induction equation
8.1 Induction by a moving conductor
8.2 Maxwell’s equations of Electromagnetism
8.3 Approximations and moving reference frames
8.4 The Induction Equation
8.5 Summary
Appendix: Ohm’s law from macroscopic considerations
8.5.1 The Magnetic Induction Equation
• Recall the equations governing electrodynamics under the
MHD approximation:
∇·B =0
∂B
∂t
∇ × B = µ0 J
∇×E =−
J = σ(E + u × B)
(1)
(2)
(3)
(4)
• Substituting from (4) into (3) gives
1
(∇ × B) = E + u × B
µ0 σ
(5)
• Take the curl of this and using the magnetic diffusivity η =
∇ × (η∇ × B) = ∇ × E + ∇ × (u × B) (6)
1
µ0 σ
8.5.1 The Magnetic Induction Equation
• Next, substituting from (2) into (6) and rearranging,
∂B
= ∇ × (u × B) − ∇ × (η∇ × B)
∂t
(7)
• If η = constant then, we can use a standard vector calculus
identity together with (1) to re-write the last term as
∇ × (η∇ × B) = η∇ × (∇ × B) = η(∇(∇ · B) − ∇2 B) = −η∇2 B
• Substituting this into (7) we arrive at the Magnetic Induction eqn:
∂B
= ∇ × (u × B) + η∇2 B
∂t
(8)
• Thus, under the MHD approximation, if we know the motion of the
conductor u and the present magnetic field, we can calculate
how the field evolves in time.
• This single equation describes the electrodynamics of Earth’s core !
8.5.2 Magnetic Reynolds Number
∂B
= ∇ × (u × B) + η∇2 B
∂t
• Assume that the velocity field has a characteristic magnitude U
• Assume that the magnetic field has a characteristic magnitude
• Assume that the lengthscale over which both fields change is
B
L
• Then the ratio of the magnitudes of the terms on the RHS will be:
UB/L
UL
|∇ × (u × B)|
∼
∼
= Rm
|η∇2 B|
ηB/L2
η
•
Rm
is known as the magnetic Reynolds number.
• For global
UL
= ULσµ0 ∼ 3 × 10−4 · 3.481 × 106 · 0.5 × 106 · 4π × 10−7
motions in : Rm =
η
∼ 650
Earth’s core
8.5.3 Perfect Conductivity: Frozen Flux
• Consider the case in which Rm = ∞ so the second term on the
right hand side is negligible (i.e. perfect conductivity), then
∂B
= ∇ × (u × B)
∂t
(9)
• Using the standard vector calculus relation,
∇ × (u × B) = (B · ∇)u − (u · ∇)B
since
∇·u=0
and ∇ · B = 0.
• So,
∂B
+ (u · ∇)B = (B · ∇)u
∂t
Advection of Magnetic
Field along with flow
or
DB
= (B · ∇)u
Dt
(10)
Stretching of magnetic
field by shear of flow
8.5.3 Perfect Conductivity: Frozen Flux
• Consider a short line element dl drawn into a fluid at some instant.
• It subsequently moves with the fluid, so the rate of change of dl
is u(r + dl) − u(r) where r and r + dl are the position vectors
at the ends of dl. The equation of evolution of dl is then:
D(dl)
= u(r + dl) − u(r) = (dl · ∇)u
Dt
• But this is identical to (10) therefore when Rm = ∞ , the
magnetic field evolves just like the line element, moving along with
the fluid.
• The magnetic field lines can be thought of as being frozen into
the fluid (i.e. fluid elements lying on a field line at some instant
must continue to lie on the fluid elements at all later times).
8.5.3 Perfect Conductivity: Frozen Flux
• In the Frozen Flux approximation, we schematically imagine flow
sweeping the magnetic field along with it, moving and stretching it.
(From Baumjohann and Treumann, 1997)
8.5.3 Perfect Conductivity: Frozen Flux
• In the limit of a perfect conductor , there are also strong
consequences for changes in the flux through material contours.
e.g. Consider a surface S bounded by a closed material contour C
that moves along with the fluid.
d
dt
!
"
"
∂B
∂B
· dS +
B · u × dl =
· dS −
u × B · dl
B · dS =
S ∂t
C
C
S
S ∂t
#
! "
∂B
=
− ∇ × (u × B) · dS = 0
∂t
S
!
!
i.e. The total flux enclosed by a material surface cannot change
with time even if the shape of the evolves : FLUX IS FROZEN !
t=t1
t=t2
c
c
• The limit Rm → ∞ is often called the frozen flux approximation.
8.5.4 Magnetic diffusion
• Next consider the opposite limit in when Rm = 0 so the first
term on the RHS is negligible (i.e.no flow), then
∂B
= η∇2 B
∂t
• This is a classic vector diffusion equation. Simple dimensional
analysis yields,
L2
ηB
B
= 2 so τD =
τD
L
η
• Thus magnetic field features with larger spatial gradients (smaller
L ) will diffuse faster for an particular magnetic diffusivity η .
• For global
scale fields in
Earth’s core :
τD = (3.481 × 106 )2 · 0.5 × 106 · 4π × 10−7 ∼ 7.6 × 1012 s
∼ 240, 000 yrs
( More precise calculations give ~ 30,000yrs for dipole part of geomagnetic field)
8.5.4 Magnetic Diffusion
• Magnetic diffusion can also allow magnetic field structures to
merge as well as decay, for example
(From Baumjohann and Treumannn, 1997)
8.5.4 Combination of Advection and Diffusion
• First, in cartesian co-ordinates (x,y,z), consider an initially constant
magnetic field B 0 = (0, 0, B0 ) only in the z direction. This is acted
on by a constant flow in the x direction which has a shear in the
z direction u = (u(z), 0, 0). Consider 1st only advection and use:
∇ × (u × B) = (B · ∇)u − (u · ∇)B
e.g.
B0
u
Integrating the last w.r.t time
and using Bx =0 at t=0 gives,
since
∇·u=0
and ∇ · B = 0.
∂Bz
= 0 => Bz = B0 for all t
∂t
∂By
= 0 => By = 0 for all t
∂t
∂u(z)
∂Bx
= B0
∂t
∂z
Bx = B0
du
t
dz
• So this simple shear causes the field to grow linearly with time in
the direction of flow (will be important for dynamo action.... ).
8.5.4 Combination of Advection and Diffusion
• But have ignored diffusion, if this were present the field growth
would be slower since the x-component of the induction equation is:
∂ 2 Bx
∂u
∂Bx
= B0
+η
∂t
∂z
∂z 2
• Diffusion will become more important as the field gradients grow,
until eventually a steady state is reach where advection and
diffusion are in balance:
∂ 2 Bx
∂u
B0
= −η
∂z
∂z 2
• Integrating twice w.r.t. z the steady state profile for Bx can be found:
!z
B0
Bx = − u(z ! )dz ! + cz + d
η
0
• So, even if diffusion is not initially important in the induction
equation, it will often be a vital ingredient of the saturated state.
Lecture 8: Fundamentals of Electromagnetism
and the Induction equation
8.1 Induction by a moving conductor
8.2 Maxwell’s equations of Electromagnetism
8.3 Approximations and moving reference frames
8.4 The Induction Equation
8.5 Summary
Appendix: Ohm’s law from microscopic considerations
8.6 Summary: self-assessment questions
(1) Can you state and describe the physical meaning of
each of Maxwell’s equations?
(2) Can you derive the magnetic Induction equation?
(3) Do you understand the consequences of the frozen flux
approximation to the Induction equation?
(4) Can you estimate the magnetic diffusion time for a
particular system?
Next time:
Dynamo Theory
References
- Baumjohann, W.A. and Treumann, R.A., (1997) Basic Space Plasma Physics,
Imperial College Press. (Detailed derivation of Generalised Ohm’s law, Section 7.3).
- Davidson, P.A., (2001) An introduction to magnetohydrodynamics, Cambridge
University Press. (Good for physical understanding, especially Chapters 1,2,4).
- Jackson, J.D., (1998) Classical Electrodynamics, John Wiley and Sons Inc.
(A wealth of detail on electromagnetism in general).
- Roberts, P.H., (2007) Theory of the geodynamo. In Treatise on Geophysics, Vol 5
Geomagnetism, Ed. M. Kono, Chapter 8.03, pp.67-102. (especially section 8.03.2)
Lecture 8: Fundamentals of Electromagnetism
and the Induction equation
8.1 Induction by a moving conductor
8.2 Maxwell’s equations of Electromagnetism
8.3 Approximations and moving reference frames
8.4 The Induction Equation
8.5 Summary
Appendix: Ohm’s law from microscopic considerations
Ohm’s Law: A microscopic derivation
• In previous section we ignored all microscopic effects and stated
Ohm’s law as an empirical result.
• But, it can also be derived from microscopic considerations of
motions of electrons and ions in a conductor.
• The equations of conservation of momentum density for the
electrons and ions are (e.g. Baumjohann and Treumann, 1997 p.140):
1
ne e
R
∂(ne v e )
+ ∇ · (ne v e v e ) = −
∇ · Pe −
(E + v e × B) +
∂t
me
me
me
1
ni e
R
∂(ni v i )
+ ∇ · (ni v i v i ) = − ∇ · P i +
(E + v i × B) +
∂t
mi
mi
mi
Temporal variation
of flux density
NL momentum
density flux
ns
ms
= number density of species
vs
= bulk velocity of species
Fluid Pressure tensor
EM Lorentz force
Collisions btw ions
and electrons
= mass of species
Ohm’s Law: A microscopic derivation
• Multiply the electron eqn by mi and the ion equation by me and
subtract the equations.
• Then assuming me /mi << 1 and ne ∼ ni ∼ n (quasi-neutral)
and defining the macroscopic current as J = e(ni v i − ne v e ) :
me ∂J
= ∇ · P e + ne(E + v e × B) − R
e ∂t
• Note the dependence of the current density only on the electron
fluid pressure, electron velocity and electron-ion collisions.
• Again assuming me /mi << 1 and ne ∼ ni ∼ n (quasi-neutral)
then the bulk fluid velocity is approximately that of the ions so,
vi ∼ u
and
ve ∼ u −
J
ne
• This allows the Lorentz force on the electrons to be split into 2
terms, one involving the bulk velocity and one the current density:
me ∂J
= ∇ · P e + ne(E + u × B) − J × B − R
e ∂t
Ohm’s Law: A microscopic derivation
• Next, we note that we can write the collision term as
J
ne2
R = nme ωc (v i − v e ) = ne
where σ =
σ
me ω c
ω
where c is the frequency of collisions.
• Using this leads to an expression of the Generalised Ohm’s law:
J
1
1
me ∂J
+
=
∇
·
P
+
(E
+
u
×
B)
−
(J × B)
e
ne2 ∂t
σ
ne
ne
pe
<< 1 can neglect pressure gradient on RHS.
ne U BLp
me σ
<< 1 can neglect current changes on LHS.
• When
ne2 τ
• When
• When
σB
<< 1
ne
can neglect Lorentz (Hall) term on RHS.
• These are all true in Earth’s core so finally we are left with,
J = σ(E + u × B)